Primary Maths Challenge toolkit

Two

HintIt is the smallest prime of all, and it divides into every single even number you can name.

The whyA prime has exactly two factors: itself and 1. Two qualifies, because its factors are 1 and 2. One does not, since it has only a single factor — which is why 1 is never counted as a prime.

The mode

HintA shop talks about its most popular size; this average does the same job for a list of numbers.

The whyIt is the only average that works on things that are not numbers at all — the most common colour, the most common shoe size, the most common answer. A set can have two of them, or none, which is exactly the kind of small print a Challenge question likes to hang a puzzle on.

The perimeter

HintIt is what a fence round a field measures, not how much grass is inside it.

The whyIt is a length, so it is measured in centimetres or metres, while the covering inside is measured in square centimetres or square metres. Two shapes can share one and differ wildly in the other: a long thin rectangle and a near-square can go all the way round in the same distance yet hold very different amounts of space. Challenge questions love that pair of facts.

180

HintTear the three corners off a paper triangle, lay them side by side, and they make a perfectly straight line.

The whyIt holds for every triangle — tall, squashed, right-angled or not — which is why it is the first thing to reach for when a question hides one angle. It also explains the special cases worth knowing: an equilateral triangle shares the total equally three ways, and the two base angles of an isosceles triangle are always the same as each other.

12

HintCount the ones round the top, then the ones round the bottom, then the uprights joining the two.

The whyFour round the top, four round the bottom, four uprights. Holding those three numbers ready saves a whole minute when a question asks about painting a cube, cutting one up, or rolling one along a table.

median

HintA motorway has a strip of the same name running down its centre, separating the two halves of the traffic.

The whyWhen there is an even number of values there is no single one in the middle, so you take the two nearest the centre and find the value halfway between them. It is the average that shrugs off an extreme reading: one enormous value drags the mean upwards but hardly moves this one at all.

366

HintIt is the year in which February gets its twenty-ninth date.

The whyA leap year is usually one that divides by 4 — but a century year only counts if it divides by 400, which is why 2000 was a leap year and 1900 was not. Date puzzles turn up in the Challenge every year, and so does the rhyme for the month lengths.

100; 60; 24

HintThe first jump is the tidy one our measuring system is built on. The other two come from clocks and calendars, which were never metric.

The whyThe first is a power of ten, so converting is only a matter of moving the digits along. Time refuses to join in: it was counted in sixties and twelves thousands of years before the metric system existed, which is why 1.5 hours is 90 minutes and not 150. Questions that mix a length with a duration are relying on exactly that trap.

The multiplication

HintBrackets and indices are dealt with first of all; adding and subtracting wait until last.

The whyThe agreed order is brackets, indices, then dividing and multiplying, then adding and subtracting — so 3 + 4 × 5 is 23, not 35. It is a convention, not a discovery: everyone agrees to read it the same way so that a written sum means one thing and not two. Calculators follow it, which is worth remembering when the answer on the screen is not the one you expected.

The square numbers

HintFor these, one of the pairs collapses into a single value, because both partners in it are identical — think of 5 times 5.

The whyTake 25: its factor pairs are 1 × 25 and 5 × 5, so the list is 1, 5, 25 — three factors, not four, because 5 is only written once. Every other number pairs up neatly. This is the fact behind the classic locker puzzle, where the doors left open at the end are exactly the ones with those numbers, and it is worth recognising the moment a question starts counting factors.

The lowest common multiple

HintIt is the first place where the two times tables meet each other again.

The whyCount on in sixes and in eights and the two lists first agree at 24, so the buses leave together every 24 minutes — not every 48. Multiplying the two numbers together always gives a moment when they coincide, but it is rarely the first one. Repeating events like buses, blinking lights and lapping runners are Challenge favourites, and they are all this same question in disguise.

No answer options given

HintFor those, nothing is offered to pick from, so a lucky stab in the dark is worth far less than it was earlier on.

The whyThe Primary Maths Challenge is set by the Mathematical Association for pupils aged 9 to 11: twenty-five questions in forty-five minutes, the first twenty multiple choice and arranged so they get harder as you go, the last five written out in full. The First Round runs in November — 9 to 20 November in 2026 — and high scorers are invited to the Bonus Round the following February. Gold, silver, bronze and participation certificates come in every pack of ten papers. The questions lean on logical reasoning rather than on formal method, so the quickest route through is very often to think rather than to calculate.